Quantity A |
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | Quantity B |

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | ||

$x^2-4y^2$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $8$ |

- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 2 of Section 6 of Practice Test 1. Those questions testing our knowledge of **Operations with Algebraic Expressions** can be kind of tricky, but never fear, PrepScholar has got your back!

## Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

Looks like we have an equation with **algebraic expressions**, then we’re comparing a quantity with algebraic expressions to a certain value. So this question likely draws on our knowledge of **Operations with Algebraic Expressions**. We also notice that Quantity A contains variables raised to **exponents $(^2)$**, which might draw on our knowledge of **Exponents and Roots**. Let’s keep what we’ve learned about these skills at the tip of our minds as we approach this question.

## What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

- We have an equation with algebraic expressions containing $x$ and $y$
- We want to compare an algebraic expression containing exponents to a certain value

## Develop a Plan

Part of becoming good at math is learning to recognize mathematical relationships that we’ve seen before, but are disguised as being different. And a key aspect to this is learning how to recognize coincidental numerical and algebraic relationships.

Looking at the equation given in this question, it is just a *tiny* bit curious how the two algebraic expressions look eerily similar. In fact, they’re virtually the same! Save for the sign change, $(x-2y)$ looks almost identical to $(x+2y)$. Definitely too much of a coincidence to ignore, that’s for sure.

Looking at Quantity A, we notice that we have variables raised to the power $2$. Ah! Now we remember a specific formula we learned when we learned about operations with algebraic expressions. Specifically:

$$(a+b)(a-b)=a^2-b^2$$

Excellent! So it looks like when we have the product of the sum of two things and their difference, the result is the difference in their squares. Let’s see if we can use this to simplify the left side of our equation.

$$(x-2y)(x+2y)=(x)^2-(2y)^2$$

Well, if we distribute that $2$ exponent inside the parentheses, we’ll get:

$$(x-2y)(x+2y)=x^2-4y^2$$

Well what do you know! This is the *exact* same as Quantity A! Strange how that worked out so nicely, right?

## Solve the Question

We know that $(x-2y)(x+2y)=4$ and that $(x-2y)(x+2y)=x^2-4y^2$. Thus, Quantity A MUST be equal to $4$, which is less than Quantity B. **The correct answer is B, Quantity B is greater**.

## What Did We Learn

Coincidental mathematical relationships form an integral part of the Quantitative section of the GRE. We should *always* be on the lookout for numbers that just seem too coincidental to be placed in a question by accident. For example, having the *exact* same expression, save for a sign change.

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